Chemistry Reference
In-Depth Information
TABLE 1.1 Crystal Systems and Bravais Lattices
The Seven
Crystal Systems
Bravais
Lattices
Lattice
Symbol
Unit Cell
Minimum Symmetry
a¼b¼g ¼ 90
Cubic
Four threefold rotation
axes at 109
23
0
to
each other
Simple
P
a ¼b ¼ c
Body centered
I
Face centered
F
a¼b¼g ¼ 90
Tetragonal
One fourfold rotation
axis or one fourfold
rotation-inversion
axis
Simple
P
a ¼b 6¼ c
Body centered
I
Hexagonal
a¼b¼90
One sixfold rotation
axis or one sixfold
axis rotation-inver-
sion axis
Simple
P
g ¼120
a ¼b 6¼ c
a¼b¼g 6¼ 90
Rhombohedral
(trigonal)
One threefold rotation
axis
Simple
R
a ¼b ¼ c
a¼b¼g ¼ 90
Orthorhombic
Any combination
of three mutually
perpendicular two-
fold rotation axes or
planes of symmetry
Simple
P
a 6¼b 6¼ c
Body centered
I
Base centered
C
Face centered
F
a¼g ¼90
Monoclinic
One twofold rotation
axis or one twofold
rotation-inversion
axis
Simple
P
b 6¼90
Base centered
C
a 6¼b 6¼ c
a 6¼b 6¼g 6¼ 90
Triclinic
None
Simple
P
a 6¼b 6¼ c
coordinates (u
0
, v
0
, w
0
) of any point on the line. The coordinates (u
0
, v
0
, w
0
) are not
necessarily integers. However, by convention, (u
0
, v
0
, w
0
) are multiplied by the
smallest number that produces integers u, v, and w. The crystal direction is described
by putting the three integers in square brackets as [uvw]. [uvw] are the indices
of a specific crystal direction; each of the indices can take a value of positive or
negative integer. All directions in a crystal that are symmetry equivalent to [uvw] are
represented by a notation with the integers in angular brackets as huvwi. For example,
in a cubic crystal all diagonals of the unit cell are symmetry equivalent. So all the
directions [111], ½111, ½111, ½111, ½111, ½111, ½111, and ½111 can be represented
by h111i. The bar over the number is for negative indices. Figure 1.4(a) shows some
lattice directions and their indices in a unit cell.
The orientation of lattice planes can be described by using a set of three integers
referred to as Miller indices. Miller indices are the reciprocal intercepts of the plane
on the unit cell axes. If the crystal planemakes fractional intercepts of 1/h,1/k,1/l with
